Exciting news! Jie Liu has proven a conjecture of mine -- thereby resolving an old conjecture of Sommese. I'll briefly explain his result in this post.

The conjecture was:

**Conjecture** (L---, now proven by Jie Liu). Let \(X\) be a smooth projective variety over \(\mathbb{C}\), and \(\mathscr{E}\) an ample vector bundle on \(X\). Suppose that $$\text{Hom}(\mathscr{E}, T_X)\not=0.$$ Then \(X\) is isomorphic to \(\mathbb{P}^n\) for some \(n\).

This conjecture was known (due to Andreatta-Wisniewski) in the case that there was a map \(\mathscr{E}\to T_X\) of constant rank; Liu in fact proves something slightly stronger:

**Theorem (Liu).** Let \(X\) be a smooth projective variety over \(\mathbb{C}\), and suppose \(T_X\) contains an ample subsheaf \(\mathscr{F}\). Then \(X\simeq \mathbb{P}^n\) for some \(n\).

The proof is quite nice. Old results of Araujo, Druel, and Kovacs immediately show that there exists an open subscheme \(X_0\subset X\) and a map \(X_0\to T\) whose fibers are isomorphic to \(\mathbb{P}^r\) for some \(r\). We wish to show that \(T\) is a point. (Up to this point, this more or less follows Andreatta-Wisniewski). This is not particularly hard if the \(\mathbb{P}^r\)-bundle \(X_0\to T\) is defined away from codimension two; indeed, in this case, one may find a proper curve in \(T\) and obtain a contradiction using standard techniques originally due to Peternell-Campana in this case -- this is essentially how Andreatta-Wisniewski proceed. The key fact is that if \(f: Y\to C\) is a \(\mathbb{P}^r\)-bundle over a proper curve \(C\), the relative tangent sheaf \(T_f\) contains no ample subsheaves.

Unfortunately if \(\mathscr{F}\) is not locally free, I don't see how to extend the fibration to codimension two, so this approach seems to be sunk.

Liu cleverly gets around this issue by working on an *open* subset of \(X\) rather than a closed subset. Unfortunately the paper is quite densely written and I do not as yet understand all the details -- if someone would like to explain what is happening "morally," I would love to hear it.

In any case, the results of the paper in which I made this conjecture allow one to deduce from this result an old conjecture of Sommese, which classifies smooth varieties containing a projective bundle as an ample divisor. This conjecture has attracted a lot of interest over the last few decades, so it's very gratifying to see it put to rest.